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Switching Kalman filter

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Type of mathematical filter

The switching Kalman filtering (SKF) method is a variant of the Kalman filter. In its generalised form, it is often attributed to Kevin P. Murphy, but related switching state-space models have been in use.

Applications

Applications of the switching Kalman filter include: Brain–computer interfaces and neural decoding, real-time decoding for continuous neural-prosthetic control, and sensorimotor learning in humans. It also has application in econometrics, signal processing, tracking, computer vision, etc. It is an alternative to the Kalman filter when the system's state has a discrete component. The additional error when using a Kalman filter instead of a Switching Kalman filter may be quantified in terms of the switching system's parameters. For example, when an industrial plant has "multiple discrete modes of behaviour, each of which having a linear (Gaussian) dynamics".

Model

There are several variants of SKF discussed in.

Special case

In the simpler case, switching state-space models are defined based on a switching variable which evolves independent of the hidden variable. The probabilistic model of such variant of SKF is as the following:

Pr ( { S t , X t ( 1 ) , … , X t ( M ) , Y t } ) = Pr ( S 1 ) ∏ t = 2 T Pr ( S t ∣ S t − 1 ) × ∏ m = 1 M Pr ( X 1 ( m ) ) ∏ t = 2 T Pr ( X t ( m ) ∣ X t − 1 ( m ) ) × ∏ t = 1 T Pr ( Y t ∣ X t ( 1 ) , … , X t ( M ) , S t ) . {\displaystyle {\begin{aligned}&\Pr(\{S_{t},X_{t}^{(1)},\ldots ,X_{t}^{(M)},Y_{t}\})\\={}&\Pr(S_{1})\prod _{t=2}^{T}\Pr(S_{t}\mid S_{t-1})\times \prod _{m=1}^{M}\Pr(X_{1}^{(m)})\prod _{t=2}^{T}\Pr(X_{t}^{(m)}\mid X_{t-1}^{(m)})\times \prod _{t=1}^{T}\Pr(Y_{t}\mid X_{t}^{(1)},\ldots ,X_{t}^{(M)},S_{t}).\end{aligned}}}

The hidden variables include not only the continuous X {\displaystyle X} , but also a discrete *switch* (or switching) variable S t {\displaystyle S_{t}} . The dynamics of the switch variable are defined by the term Pr ( S t ∣ S t − 1 ) {\displaystyle \Pr(S_{t}\mid S_{t-1})} . The probability model of X {\displaystyle X} and Y {\displaystyle Y} can depend on S t {\displaystyle S_{t}} .

The switch variable can take its values from a set S t ∈ { 1 , 2 , … , M } {\displaystyle S_{t}\in \{1,2,\ldots ,M\}} . This changes the joint distribution ( X t , Y t ) {\displaystyle (X_{t},Y_{t})} which is a separate multivariate Gaussian distribution in case of each value of S t {\displaystyle S_{t}} .

General case

In more generalised variants, the switch variable affects the dynamics of X t {\displaystyle X_{t}} , e.g. through Pr ( X t ∣ X t − 1 , S t ) {\displaystyle \Pr(X_{t}\mid X_{t-1},S_{t})} . The filtering and smoothing procedure for general cases is discussed in.

References

  1. ^ K. P. Murphy, "Switching Kalman Filters", Compaq Cambridge Research Lab Tech. Report 98-10, 1998
  2. K. Murphy. Switching Kalman filters. Technical report, U. C. Berkeley, 1998.
  3. K. Murphy. Dynamic Bayesian Networks: Representation, Inference and Learning. PhD thesis, University of California, Berkeley, Computer Science Division, 2002.
  4. Kalman Filtering and Neural Networks. Edited by Simon Haykin. ISBN 0-471-22154-6
  5. Wu, Wei, Michael J. Black, David Bryant Mumford, Yun Gao, Elie Bienenstock, and John P. Donoghue. 2004. Modelling and decoding motor cortical activity using a switching Kalman filter. IEEE Transactions on Biomedical Engineering 51(6): 933-942. doi:10.1109/TBME.2004.826666
  6. Heald JB, Ingram JN, Flanagan JR, Wolpert DM. Multiple motor memories are learned to control different points on a tool. Nature Human Behaviour. 2, 300–311, (2018).
  7. ^ Kim, C.-J. (1994). Dynamic linear models with Markov-switching. J. Econometrics, 60:1–22.
  8. ^ Bar-Shalom, Y. and Li, X.-R. (1993). Estimation and Tracking. Artech House, Boston, MA.
  9. Karimi, Parisa (2021). "Quantification of mismatch error in randomly switching linear state-space models". IEEE Signal Processing Letters. 28: 2008–2012. arXiv:2012.04542. Bibcode:2021ISPL...28.2008K. doi:10.1109/LSP.2021.3116504. S2CID 227745283.
  10. ^ Zoubin Ghahramani, Geoffrey E. Hinton. Variational Learning for Switching State-Space Models. Neural Computation, 12(4):963–996.
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